Counting stable solutions of sparse polynomial systems in chemistry

Please always quote using this URN: urn:nbn:de:0297-zib-6007

The polynomial differential system modelling the behavior of a chemical reaction is given by graphtheoretic structures. The concepts from toric geometry are applied to study the steady states and stable steady states. Deformed toric varieties give some insight and enable graph theoretic interpretations. The importance of the circuits in the directed graph are emphazised. The counting of positive solutions of a sparse polynomial system by B.\ Sturmfels is generalized to the counting of stable positive solutions in case of a polynomial differential equation. The generalization is based on a method by sparse resultants to detect whether a system may have a Hopf bifurcation. Special examples from chemistry are used to illustrate the theoretical results.

Metadaten
Author:	Karin Gatermann
Document Type:	ZIB-Report
Tag:	circuit; mass action kinetics; stability; toric variety
MSC-Classification:	05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
	14-XX ALGEBRAIC GEOMETRY / 14Mxx Special varieties / 14M25 Toric varieties, Newton polyhedra [See also 52B20]
	34-XX ORDINARY DIFFERENTIAL EQUATIONS / 34Dxx Stability theory [See also 37C75, 93Dxx] / 34D99 None of the above, but in this section
	68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area) / 68Wxx Algorithms (For numerical algorithms, see 65-XX; for combinatorics and graph theory, see 05C85, 68Rxx) / 68W30 Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]
Date of first Publication:	2000/09/15
Series (Serial Number):	ZIB-Report (00-32)
ZIB-Reportnumber:	00-32
Published in:	Appeared in: Contemporary Mathematics, vol. 286, Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Eds. E. Green, S. Hosten, R. Laubenbacher, V. Powers. AMS, 2001, pp. 53-69

Open Access